I assume you are essentially renting a room in the house in which they also live with you and your partner and baby? I do not read your post as you renting their house off them and having the whole house to yourself.
£450-550 per month is the going rate for room rentals where I live in the Northeast. Then all the tenants split the bills.
Here are my bills:
Internet £36 per month
Council tax £2.9k a year - will go past 3k in April
Water £726 per year - until March it was £612
Energy is absolutely horrific at the moment at around £470 a month
Insurance £330 per year - that’s just for contents insurance - I live on a river. Homeowners have to pay for both contents and buildings insurance to cover the cost of rebuilding the whole house if the worst was to happen to it. Mine is particularly expensive due to the river location. My last contents not on a river last year was £45 a year.
TV licence £15 per month
My rent - for the whole house, not just a room - £1650 per month
This is for a four bedroom bungalow.
Running a household is painful these days.
I hope that gives you some vague idea of bills.
A two bedroom terrace has a council tax of £1515 ish per year where I am - it does get cheaper the smaller the house is (as does water, but it doesn’t just halve).
When I turned 16 and got my first job, my mum charged me £200 a month from my £550 take home - that included food - but was just for one person, not three. So if you are being charged £550 from an income of £1800 I would say your partner’s family are being quite generous and probably just literally trying to cover their costs whilst at the same time leaving you with enough money to save up to get some independence.
And to give you an idea of how much prices have risen - my energy is 50% more than in 2022 and my 2021-2022 water bill was £235 instead of £726.
Good luck with the arrangement. I hope it works out well for you all.